Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $k = \dfrac{p^3 + 5p^2 - 6p}{4p^3 + 36p^2 + 72p} \times \dfrac{-2p - 6}{p + 1} $
Answer: First factor out any common factors. $k = \dfrac{p(p^2 + 5p - 6)}{4p(p^2 + 9p + 18)} \times \dfrac{-2(p + 3)}{p + 1} $ Then factor the quadratic expressions. $k = \dfrac {p(p + 6)(p - 1)} {4p(p + 6)(p + 3)} \times \dfrac {-2(p + 3)} {p + 1} $ Then multiply the two numerators and multiply the two denominators. $k = \dfrac { p(p + 6)(p - 1) \times -2(p + 3)} { 4p(p + 6)(p + 3) \times (p + 1)} $ $k = \dfrac {-2p(p + 6)(p - 1)(p + 3)} {4p(p + 6)(p + 3)(p + 1)} $ Notice that $(p + 6)$ and $(p + 3)$ appear in both the numerator and denominator so we can cancel them. $k = \dfrac {-2p\cancel{(p + 6)}(p - 1)(p + 3)} {4p\cancel{(p + 6)}(p + 3)(p + 1)} $ We are dividing by $p + 6$ , so $p + 6 \neq 0$ Therefore, $p \neq -6$ $k = \dfrac {-2p\cancel{(p + 6)}(p - 1)\cancel{(p + 3)}} {4p\cancel{(p + 6)}\cancel{(p + 3)}(p + 1)} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $k = \dfrac {-2p(p - 1)} {4p(p + 1)} $ $ k = \dfrac{-(p - 1)}{2(p + 1)}; p \neq -6; p \neq -3 $